What is the integral of arctan(3t) ?

The integral of arctan(3t) is tarctan(3t)-1/6 ln|9t^2+1|+C

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integral of arctan(3t)

\int arctan (3 t) d t

t arctan (3 t) - \frac{1}{6} ln 9 t^{2} + 1 + C

Solution steps

\int arctan (3 t) d t

Apply Integration By Parts

= t arctan (3 t) - \int \frac{3 t}{9 t ^{2} + 1} d t

\int \frac{3 t}{9 t ^{2} + 1} d t = \frac{1}{6} ln 9 t^{2} + 1

= t arctan (3 t) - \frac{1}{6} ln 9 t^{2} + 1

Add a constant to the solution

= t arctan (3 t) - \frac{1}{6} ln 9 t^{2} + 1 + C

What is the integral of arctan(3t) ?
The integral of arctan(3t) is tarctan(3t)-1/6 ln|9t^2+1|+C